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Supercuspidal representations: An exhaustion theorem

Author(s): Ju-Lee Kim
Journal: J. Amer. Math. Soc. 20 (2007), 273-320.
MSC (2000): Primary 22E50; Secondary 22E35, 20G25.
Posted: October 24, 2006
MathSciNet review: 2276772
Retrieve article in: PDF

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Abstract: Let be a reductive -adic group. We prove that all supercuspidal representations of arise through Yu's construction subject to certain hypotheses on (depending on ). As a corollary, under the same hypotheses, we see that any supercuspidal representation is compactly induced from a representation of an open subgroup which is compact modulo the center.

References:

1.: J. Adler, Refined anisotropic -types and supercuspidal representations, Pacific J. Math. 185 (1998), 1-32. MR 1653184 (2000f:22019)
2.: J. Adler and S. DeBacker, Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive -adic group, Michigan J. Math. 50 (2002), no. 2, 263-286. MR 1914065 (2003g:22016)
3.: -, Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups, J. Reine Angew. Math. 575 (2004), 1-35. MR 2097545 (2005j:22008)
4.: J. Adler and A. Roche, An intertwining result for -adic groups, Canad. J. Math. 52 (2000), no. 3, 449-467. MR 1758228 (2001m:22032)
5.: C. Bushnell and P. Kutzko, The admissible dual of via compact open subgroups, 129, Annals of Math Studeis, Princeton Univ. Press, 1993. MR 1204652 (94h:22007)
6.: -, The admissible dual of I, 26, Ann. Sci. ENS, 261-280. MR 1209709 (94a:22033)
7.: -, The admissible dual of II, Proc. London Math. Soc. 68 (1994), 317-379. MR 1253507 (94k:22035)
8.: -, Smooth representations of reductive -adic groups: structure theory via types, Proc. London Math. Soc. 77 (1998), no. 3, 582-634. MR 1643417 (2000c:22014)
9.: F. Bruhat and J. Tits, Groupes réductifs sur un corps local I, Publ. Math. I.H.E.S. 41 (1972), 5-251. MR 0327923 (48:6265)
10.: W. Casselman, Introduction to the theory of admissible representation of -adic reductive groups, preprint.
11.: S. DeBacker, Homogeneity results for invariant distributions of a reductive -adic group, Ann. Sci. École Norm. Sup. (4) 35 (2002), 391-422. MR 1914003 (2003i:22019)
12.: -, Parameterizing nilpotent orbits via Bruhat-Tits theory, Ann. of Math. 156 (2002), 295-331. MR 1935848 (2003i:20086)
13.: J. Dixmier, Les -algèbres et leurs représentations, Éditions Jacques Gabay, Paris, 1996. MR 1452364 (98a:46066)
14.: J. Hakim and F. Murnaghan, Distinguished tame supercuspidal representations, work in progress.
15.: Harish-Chandra, Admissible invariant distributions on reductive -adic groups, Preface and notes by Stephen DeBacker and Paul J. Sally, Jr., University Lecture Series, 16, American Mathematical Society, Providence, RI, 1999. MR 1702257 (2001b:22015)
16.: -, Eisenstein series over finite fields, Collected papers, Springer Verlag, New York, 1984. MR 0457579 (56:15784)
17.: -, The Plancherel formula for reductive -adic groups, Collected Papers, 4, Springer-Verlag, Berlin, 1976 MR 0726026 (85e:01061d)
18.: R. Howe, The Fourier transform and germs of characters (case of Gl over a -adic field), Math. Ann. 208 (1974), 305-322.MR 0342645 (49:7391)
19.: -, Some qualitative results on the representation theory of over a -adic field, Pacific J. of Math. 73 (1977), 479-538.MR 0492088 (58:11242)
20.: -, Tamely ramified supercuspidal representations of , Pacific J. of Math. 73 (1977), 437-460. MR 0492087 (58:11241)
21.: R. Howe with collaboration of A. Moy, Harish Chandra Homomorphisms for -adic groups, vol. 59, CBMS regional conference series, AMS, Providence, RI, 1985.MR 0821216 (87h:22023)
22.: R. Howe and A. Moy, Hecke algebra isomorphisms for over a p-adic field, J. Algebra 131 (1990), 388-424. MR 1058553 (91f:22031)
23.: R. Howlett and G. Lehrer, On Harish-Chandra induction and restriction for modules of Levi subgroups, J. Algebra 165 (1994), 172-183.MR 1272585 (95d:20025)
24.: J. L. Kim, Hecke algebras of classical groups over -adic fields and supercuspidal representations, Amer. J. Math. 121 (1999), 967-1029.MR 1713299 (2001g:11075)
25.: -, Dual blobs and Plancherel formulas, Bull. Soc. Math. France 132 (2004), no. 1, 55-80. MR 2075916 (2005f:22025)
26.: J. L. Kim and F. Murnaghan, Character expansions and unrefined minimal ${\mathsf K}$ -types, Amer. J. Math. 125 (2003), 1199-1234.MR 2018660 (2004k:22024)
27.: -, ${\mathsf K}$ -types and $\Gamma$ -asymptotic expansions, J. Reine Angew. Math. 592 (2006), 189-236. MR 2222734
28.: L. Morris, Level zero -types, Compositio Math. 118 (1999), 135-157.MR 1713308 (2000g:22029)
29.: A. Moy, Local constants and the tame Langlands correspondence, Amer. J. Math. 108 (1986), 863-930. MR 0853218 (88b:11081)
30.: A. Moy and G. Prasad, Unrefined minimal -types for -adic groups, Inv. Math. 116 (1994), 393-408. MR 1253198 (95f:22023)
31.: -, Jacquet functors and unrefined minimal -types, Comment. Math. Helvetici 71 (1996), 98-121. MR 1371680 (97c:22021)
32.: G. Prasad, Galois fixed points in the Bruhat-Tits buildings of a reductive group, Bulletin Soc. Math. France 129 (2001), no. 2, 169-174.MR 1871292 (2002j:20088)
33.: G. Rousseau, Immeubles des groupes réductifs sur les corps locaux, Thèse, Paris XI (1977).MR 0491992 (58:11158)
34.: S. Stevens, Semisimple characters for -adic classical groups, Duke Math. J. 127 (2005), no. 1, 123-173. MR 2126498 (2006a:22017)
35.: M. Tadic, Geometry of dual spaces of reductive groups, J. d'Analyse Math. 51 (1988), 139-181. MR 0963153 (90c:22057)
36.: J.-L. Waldspurger, Homogénéité de certaines distributions sur les groupes -adiques, Inst. Hautes Études Sci. Publ. Math. No. 81 (1995), 25-72.MR 1361755 (98f:22023)
37.: -, La formule de Plancherel pour les groupes -adiques d'après Harish-Chandra, J. Inst. Math. Jussieu 2 (2003), no. 2, 235-333.MR 1989693 (2004d:22009)
38.: J. K. Yu, Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), no. 3, 579-622. MR 1824988 (2002f:22033)

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